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Theorem sbhypf 2779
Description: Introduce an explicit substitution into an implicit substitution hypothesis. See also csbhypf . (Contributed by Raph Levien, 10-Apr-2004.)
Hypotheses
Ref Expression
sbhypf.1  |-  F/ x ps
sbhypf.2  |-  ( x  =  A  ->  ( ph 
<->  ps ) )
Assertion
Ref Expression
sbhypf  |-  ( y  =  A  ->  ( [ y  /  x ] ph  <->  ps ) )
Distinct variable groups:    x, A    x, y
Allowed substitution hints:    ph( x, y)    ps( x, y)    A( y)

Proof of Theorem sbhypf
StepHypRef Expression
1 vex 2733 . . 3  |-  y  e. 
_V
2 eqeq1 2177 . . 3  |-  ( x  =  y  ->  (
x  =  A  <->  y  =  A ) )
31, 2ceqsexv 2769 . 2  |-  ( E. x ( x  =  y  /\  x  =  A )  <->  y  =  A )
4 nfs1v 1932 . . . 4  |-  F/ x [ y  /  x ] ph
5 sbhypf.1 . . . 4  |-  F/ x ps
64, 5nfbi 1582 . . 3  |-  F/ x
( [ y  /  x ] ph  <->  ps )
7 sbequ12 1764 . . . . 5  |-  ( x  =  y  ->  ( ph 
<->  [ y  /  x ] ph ) )
87bicomd 140 . . . 4  |-  ( x  =  y  ->  ( [ y  /  x ] ph  <->  ph ) )
9 sbhypf.2 . . . 4  |-  ( x  =  A  ->  ( ph 
<->  ps ) )
108, 9sylan9bb 459 . . 3  |-  ( ( x  =  y  /\  x  =  A )  ->  ( [ y  /  x ] ph  <->  ps )
)
116, 10exlimi 1587 . 2  |-  ( E. x ( x  =  y  /\  x  =  A )  ->  ( [ y  /  x ] ph  <->  ps ) )
123, 11sylbir 134 1  |-  ( y  =  A  ->  ( [ y  /  x ] ph  <->  ps ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 103    <-> wb 104    = wceq 1348   F/wnf 1453   E.wex 1485   [wsb 1755
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-5 1440  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-11 1499  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-ext 2152
This theorem depends on definitions:  df-bi 116  df-tru 1351  df-nf 1454  df-sb 1756  df-clab 2157  df-cleq 2163  df-clel 2166  df-v 2732
This theorem is referenced by:  mob2  2910  cbvmptf  4083  tfisi  4571  ralxpf  4757  rexxpf  4758  nn0ind-raph  9329
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